Summary
We discuss a new approach to regularity theory for almost minimizers of variational integrals in geometric measure theory or in the classical calculus of variations. This method is direct, exhibiting the dependence of the regularity estimates on the structural data of the variational integrand in explicit form; it requires only weak growth and smoothness assumptions on the integrand; it allows a unified treatment of interior and boundary regularity; and it leads to new regularity results which give the best possible modulus of continuity for the derivative of the almost minimizer in a variety of situations.
Keywords
- Regularity Result
- Partial Regularity
- Boundary Regularity
- Geometric Measure Theory
- Nonlinear Elliptic System
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Allard, W.K.: On the first variation of a varifold. Annals of Math., 95, 417–491, 1972
Almgren, F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure. Annals of Math., 87, 321–391, 1968
Almgren, F.J.: Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints. Mem. Amer. Math. Soc., 165. 1976
Almgren, F.J.: Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two. Princeton Univ. 1984, 3745 g; see also Bull. Amer. Math. Soc, 8, 327–328 (1983) and Almgren’s big regularity paper, ed. Scheffer, V., Taylor, J. E., World Scientific Singapore 2000
Acerbi, E., Fusco, N.: Semicontinuity problems in the calculus of variations. Arch. Ration. Mech. Anal., 86, 125–145, 1984
Acerbi, E., Fusco, N.: A regularity theorem for minimizers of quasiconvex integrals. Arch. Ration. Mech. Anal., 99, 261–281, 1987
Anzellotti, G.: On the C 1,α-regularity of ω-minima of quadratic functionals. Boll. Unione Mat. Ital., VI. Ser., C., Anal. Funz. Appl., 2, 195–212, 1983
Anzellotti, G., Giaquinta, M.: Convex functions and partial regularity. Arch. Ration. Mech. Anal., 102. 243–272, 1988
Ball, J.M.: Convexity conditions and existence theorems in nonlinear elasticity. Arch. Ration. Mech. Anal., 63, 337–403, 1977
Bombieri, E.: Regularity theory for almost minimal currents. Arch. Ration. Mech. Anal., 7, 99–130, 1982
Campanato, S.: Equazioni ellittiche del IIe ordine e spazi. Ann. Mat. Pura Appl., 69, 321–381, 1965
Dacorogna, B.: Direct methods in the calculus of variations. Springer, Berlin-Heidelberg-New York, 1989
De Giorgi, E.: Frontiere orientate di misura minima. Seminaro Mat. Scuola Norm. Sup. Pisa, 1–56, 1961
De Giorgi, E.: Un esempio di estremali discontinue per un problema variazionale di tipo ellitico. Boll. Unione Mat. Ital., IV. Ser., 1, 135–137, 1968
Duzaar, F.: Boundary regularity for area minimizing currents with prescribed volume. J. Geom. Anal., 7, 585–592, 1997
Duzaar, F., Gastel, A.: Nonlinear elliptic systems with Dini continuous coefficients. Archiv der Math., 78, 58–73, 2002
Duzaar, F., Gastel, A., Grotowski, J.F.: Partial regularity for almost-minimizers of quasiconvex integrals. SIAM J. Math. Analysis, 32, 665–687, 2000
Duzaar, F., Gastel, A., Grotowski, J.F.: Optimal partial regularity for nonlinear elliptic systems of higher order. J. Math. Sci., Tokyo, 8, 463–499, 2001
Duzaar, F., Grotowski, J.F.: Partial regularity for nonlinear elliptic systems: the method of A-harmonic approximation. Manuscr. Math., 103. 267–298, 2000
Duzaar, F., Steffen, K.: A minimizing currents. Manuscr. Math.,80, 403–447, 1993
Duzaar, F., Steffen, K.: Boundary regularity for minimizing currents with prescribed mean curvature. Calc. Var. Partial Diff. Equ., 1, 355–406, 1993
Duzaar, F., Steffen, K.: Optimal interior and boundary regularity for almost minimal currents to elliptic integrands. J. Reine Angew. Math., 546. 73–138, 2002
Evans, L.C.: Quasiconvexity and partial regularity in the calculus of variations. Arch. Ration. Mech. Anal., 95, 227–252, 1986
Federer, H.: Geometric Measure Theory. Springer, Berlin-Heidelberg-New York, 1969
Federer, H.: The singular set of area minimizing rectifiable currents with codimension one and of area minimizing flat chains modulo two with arbitrary codimension. Bull. Amer. Math. Soc., 76, 767–771, 1970
Fusco, N., Hutchinson, J.: C 1,α partial regularity of functions minimising quasiconvex integrals. Manuscr. Math., 54, 121–143, 1985
Gehring, F.W.: The L p-integrability of the partial derivatives of a quasiconformal map. Acta Math., 130. 265–277, 1973
Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton, 1983
Giaquinta, M.: Introduction to Regularity Theory for Nonlinear Elliptic Systems. Birkhäuser, Basel-Boston-Berlin, 1993
Giaquinta, M., Ivert, P.-A.: Partial regularity for minima of variational integrals. Ark. Math., 25, 221–229, 1987
Giaquinta, M., Modica, G.: Regularity results for some classes of higher order nonlinear elliptic systems. J. Reine Angew. Math., 311/312. 125–169, 1979
Giaquinta, M. Modica, G.: Partial regularity of minimizers of quasiconvex integrals. Ann. Inst. H. Poincaré, Analyse non linéaire, 3, 185–208, 1986
Giusti, E., Miranda: Sulla regolaritlà delle soluzioni deboli di una classe di sistemi ellittici quasi-lineari. Arch. Ration. Mech. Anal., 31. 173–184, 1968
Gonzalez, E., Massari, U., Tamanini, L: On the regularity of boundaries of sets minimizing perimeter with a volume constraint. Indiana Univ. Math. J., 32, 25–37, 1983
Grotowski, J.F.: Boundary regularity for nonlinear elliptic systems. To appear in: Calc. Var. Partial Differ. Equ.
Grotowski, J.F.: Boundary regularity for quasilinear elliptic systems. To appear in: Commun. Partial Differ. Equations.
Hamburger, C: A new partial regularity proof for solutions of nonlinear elliptic equations. Manuscr. Math., 95, 11–31, 1998
Hardt, R.: On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand. Commun. Partial Diff. Equations, 2, 1163–1232, 1977
Hardt, R., Simon, L.: Boundary regularity and embedded solutions for the oriented Plateau problem. Annals of Math., 110. 439–486, 1979
Hartman, P., Wintrier, A.: On uniform Dini conditions in the theory of linear partial differential equations of elliptic type. Am. J. Math., 77, 329–354, 1955
Jost, J., Meier, M.: Boundary regularity for minima of certain quadratic functionals. Math. Ann., 262. 549–561, 1983
Kovats, J.: Fully nonlinear elliptic equations and the Dini condition. Commun. Partial Diff. Equations, 22, 1911–1927, 1997
Kronz, M.: Partial regularity results for minimizers of quasiconvex functional of higher order. Ann. Inst. H. Poincaré, Analyse non linéaire, 19. 81–112, 2002
Maz’ya, V.G.: Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients. Funkts. Anal. Prilozh., 2, 53–57 (1968); translated in Funct. Anal. Appl, 2, 230-234, 1968
Miranda, M: Frontilère orientate con ostacoli. Ann. Univ. Ferrara, 16, 29–37, 1971
Morrey, C.B.: Quasi-convexity and the lower semicontinuity of multiple integrals. Pacific J. Math., 2, 25–53, 1952
Morrey, C.B.: Multiple Integrals in the Calculus of Variations. Springer, Heidelberg-New York, 1966
Morrey, C.B.: Partial regularity results for non-linear elliptic systems. J. Math. Mech., 17, 649–670, 1968
Simon, L.: Lectures on Geometric Measure Theory. Proc. CMA, Vol. 3, ANU Canberra, 1983
Simon, L.: Theorems on Regularity and Singularity of Energy Minimizing Maps. Birkhäuser, Basel-Boston-Berlin, 1996
Schoen, R., Simon, L.:A new proof of the regularity for rectifiable currents which minimize parametric elliptic functionals. Indiana Univ. Math. J., 31. 413–43, 1982
Schoen, R., Simon, L., Almgren, F.J.: Regularity and singularity estimates on hypersurfaces minimizing parametric elliptic variational integrals. Acta Math., 139. 217–265, 1977
Tamanini, I.: Regularity results for almost minimal oriented hypersurfaces in ℝn. Quad. Dipt. Mat. Uni. Lecce, 1-1984. 1984
Tamanini, I.: Boundaries of Caccioppoli sets with Hülder-continuous normal vector. J. Reine Angew. Math., 334. 27–39, 1982
Tamanini, I.: Variational problems of least area type with constraints. Ann. Univ. Ferrara, 34. 183–217, 1988
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Duzaar, F., Grotowski, J.F., Steffen, K. (2003). Optimal Regularity Results via A-Harmonic Approximation. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-55627-2_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44051-2
Online ISBN: 978-3-642-55627-2
eBook Packages: Springer Book Archive